PRACTICAL USES OF EXISTENTIAL RULES IN KNOWLEDGE REPRESENTATION

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PRACTICAL USES OF EXISTENTIAL RULES IN KNOWLEDGE REPRESENTATION

Part 2: Solving Horn-ALCClassification with Existential Rules

David Carral,¹Markus Kr ¨otzsch,¹and Jacopo Urbani² 1. TU Dresden

2. VU University Amsterdam

Special thanks to Irina Dragoste,¹Ceriel Jacobs,²and Maximilian Marx¹ for their invaluable contributions to the software used in this tutorial

KR 2020, September 13, 2020

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Outline

Goal

Show some example where eitherrulesorrelated ideaswere crucial to achieve the state of the art

• Horn-ALCreasoning

• PLP

• Data integration

• Stream reasoning

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1 ^st Scenario: Horn- ALC reasoning

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The Description Logic Horn- ALC _{: Syntax}

Definition. A Horn-ALContology is a set of Horn-ALCaxioms:

Av ⊥ > vB AvB AuEvB ∃R.AvB Av ∀R.B Av ∃R.B In the above;A,B, andEare concept names; andRis a role name.

Remark. Note the axioms of the formAv ∀R.B, which are notEL, such as:

CheesePizzav ∀HasTopping.Cheese

The axiom states that “all toppings in a cheese pizza are cheese toppings”.

Even though Horn-ALCis not much more expressive thanEL, (Krötzsch, Rudolph, and Hitzler 2013) have showed that:

Theorem. Solving classification over Horn-ALCis ExpTime-complete.

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A Consequence-Based Calculus to Solve Classification

R^C_A

AvA :A∈Concepts(O) R⁺_∃ CvA

Cv ∃R.B :Av ∃R.B∈ O R^∃_A Cv ∃R.D D∈D

DvD R⁻_∃ Cv ∃R.D DvA

CvB :∃R.AvB∈ O R¹_u CvA

CvB :> vB∈ O R^⊥_∃ Cv ∃R.D Dv ⊥ Cv ⊥ R¹_u CvA

CvB :AvB∈ O R∀

Cv ∃R.D CvA

Cv ∃R.(DuB) :Av ∀R.B∈ O R²_u CvA CvE

CvB :AuEvB∈ O

Figure: Classification Calculus for Horn-ALC. WhereA,B, andEare concept names;R is a role name; andCandDare conjunctions of concept names

Remark. The above procedure is based on the work of (Kazakov 2009).

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Consequence-Based Calculus: Complexity

Theorem. The Horn-ALCclassification calculus runs in exponential time in the size of the input ontologyO.

Remark. Note that this calculus produces inferences of the form

(1) CvB and (2) Cv ∃R.D

whereBis a concept name,Ris a role name, andCandDare conjunctions of concept names. Therefore, on inputO, the calculus may produce at most

2^|^Concepts^(O)|× |Concepts(O)| and 2×2^|^Concepts^(O)|× |Roles(O)|

inferences of type (1) and (2), respectively.

Remark. Since classification over Horn-ALCis an ExpTime-complete problem, the calculus is worst-case optimal.

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Implementing the Classification Calculus: Datalog

Because of the following result, we can not implement the Horn-ALCclassification calculus using a fixed Datalog rule set:

Theorem. The data complexity of fact entailment over Datalog is in P.

Assume that we can implement the Horn-ALCclassification calculus with a fixed Datalog rule set (as we did for theELclassification calculus). Then:

1. By the above theorem, we could solve Horn-ALCclassification in polynomial time.

2. By (1), we could solve an ExpTime-hard problem in polynomial time.

3. By (2), P=ExpTime( )

Remark. To implement the Horn-ALCclassification calculus (or any other procedure that solves Horn-ALCclassification), we need a rule-based language with ExpTime-hard data complexity!

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Implementing the Classification Calculus: Datalog(S)

We study Datalog(S), an extension of Datalog that can model exponential computations.

Example. Consider the following Datalog(S) rule set:

Person(x)→LikesAll(x,∅) LikesAll(x,S)∧Likes(x,y)→LikesAll(x,S∪ {y})

LikesAll(x,S)→AllLikeAll({x},S) AllLikeAll(S,T)∧LikesAll(x,T)→AllLikeAll(X∪ {x},T)

AllLikeAll(S,S)∧alice∈S→CliqueOfAlice(S)

Theorem. Checking fact entailment for Datalog(S) is ExpTime-complete for both data and combined complexity.

See (Carral et al. 2019) for a complete proof of the above result.

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Implementing the Classification Calculus: Datalog(S)

Using a function to encode the axioms and entities in an input ontology as facts and a fixed Datalog(S) rule set, we can implement the Horn-ALCclassification calculus.

Example. For an ontologyO, let Facts(O)be the fact set such that:

Av ⊥ 7→nf:axiomv(cA,c⊥) ∃R.AvB7→nf:axiom∃v(cA,cR,cB)

> vB7→nf:axiomv(c>,cB) Av ∀R.B7→nf:axiomv∀(cA,cR,cB) AvB7→nf:axiomv(cA,cB) Av ∃R.B7→nf:axiomv∃(cA,cR,cB) AuEvB7→nf:axiomuv(c_A,c_E,c_B) A∈Concepts(O)7→nf:concept(c_E) In the above;cA,cB,cE,c>, andc⊥ are fresh constants unique forA,B,E,>, and

⊥, respectively; andc_R is a fresh constant uniqueR.

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Implementing the Classification Calculus: Datalog(S)

We translate the production rules in the Horn-ALCclassification calculus (left) into analogous Datalog(S) rules (right):

AvA:A∈Concepts(O) nf:concept(a)

→SC({a},a)

Cv ∃R.D

DvD :D∈D

Ex(C,r,D)∧d∈D

→SC(D,d)

CvA

CvB:> vB∈ O SC(C,a)∧nf:axiomv(c>,b)

→SC(C,b)

CvA

CvB:AvB∈ O SC(C,a)∧nf:axiomv(a,b)

→SC(C,b)

CvA CvE

CvB :AuEvB∈ O SC(C,a)∧SC(C,e)∧nf:axiomuv(a,e,b)

→SC(C,b)

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Implementing the Classification Calculus: Datalog(S)

We translate the production rules in the Horn-ALCclassification calculus (left) into analogous Datalog(S) rules (right):

CvA

Cv ∃R.B :Av ∃R.B∈ O SC(C,a)∧nf:axiomv∃(a,r,b)

→Ex(C,r,{b})

Cv ∃R.D DvA

CvB :∃R.AvB∈ O Ex(C,r,D)∧SC(D,a)∧nf:axiom_∃v(r,a,b)

→SC(C,b)

Cv ∃R.D Dv ⊥ Cv ⊥

Ex(C,r,D)∧SC(D,c⊥)

→SC(C,c_⊥)

Cv ∃R.D CvA

Cv ∃R.(DuB) :Av ∀R.B∈ O Ex(C,r,D)∧SC(C,a)∧nf:axiomv∀(a,r,b)

→Ex(C,r,D∪ {b})

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Implementing the Classification Calculus: Datalog(S)

Definition. LetR_HALCbe the rule set containing all of the above rules:

nf:concept(a)→SC({a},a) SC(C,a)∧nf:axiomv(c>,b)→SC(C,b) Ex(C,r,D)∧d∈D→SC(D,d) SC(C,a)∧nf:axiomv(a,b)→SC(C,b)

SC(C,a)∧SC(C,e)∧nf:axiom_uv(a,e,b)→SC(C,b) SC(C,a)∧nf:axiomv∃(a,r,b)→Ex(C,r,{b}) Ex(C,r,D)∧SC(D,a)∧nf:axiom∃v(r,a,b)→SC(C,b)

Ex(C,r,D)∧SC(D,c⊥)→SC(C,c⊥) Ex(C,r,D)∧SC(C,a)∧nf:axiomv∀(a,r,b)→Ex(C,r,D∪ {b})

Theorem. Consider a Horn-ALContologyOand an axiom of the formA v B.

Then,O |=AvBif and only ifR_HALC∪Facts(O)|=SC(cA,cB).

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Implementing the Classification Calculus: ∃ _-Rules

Alas, VLog does not support Datalog(S) reasoning. But there maybe some other rule-based language with ExpTime-hard data complexity that we can use!

The following result is a recent finding by (Krötzsch, Marx, and Rudolph 2019):

Theorem. The data complexity of fact entailment over rule sets that terminate with respect to the restricted chase is ExpTime-hard.

Remark. Note that the data complexity of fact entailment over existential rule sets that terminate with respect to the Skolem chase is in P.

(Carral et al. 2019) have proposed a translation from Datalog(S) into∃-rules such that:

• The resulting rule sets terminate w.r.t. theDatalog-first restricted chase.

• Fact entailment is “preserved”.

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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From Datalog(S) to Existential Rules

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Solving ExpTime-hard Problems

Step-by-step procedure to implement an ExpTime algorithm with VLog:

1. Encode input using a set of factsF.

2. Encode ExpTime algorithm using a fixed Datalog(S) rule setR.

3. Apply the translation by (Carral et al. 2019) toRto obtain a setR⁰of existential rules such that:

– The rule setR⁰“preserves” fact entailment overR.

– The rule setR⁰terminates w.r.t. the Datalog-first restricted chase.

4. Use VLog to compute all of the consequences ofR⁰∪ F.

Remark. For a detailed explanation of the above procedure, see (Carral, Dragoste, and Rudolph 2020).

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Evaluation: Classification

ID #Ax. #SC VLog Konclude

00040 223K 1051K 432s 5s

00048 142K 718K 387s 3s

00477 318K 162K 1s 3s

00533 159K 965K 132s 2s

00786 152K 2283K 549s 14s

Figure: Ontologies and results for classification showing: axiom count, number ofSC facts derived, and reasoning times for VLog and Konclude

Remark. Presented in (Carral et al. 2019).

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Evaluation: Class Retrieval

Definition. A Horn-ALContology is a set of Horn-ALCaxioms:

Av ⊥ > vB AvB AuEvB

∃R.AvB Av ∀R.B Av ∃R.B A(a) R(a,b) WhereA,B, andEare concepts;Ris a role; andaandbare individuals.

Definition. Class retrieval is the reasoning task of computing all axioms of the formA(a)that are logically entailed by some input ontologyO.

Remark. The Horn-ALCclassification calculus can be extended with 3 rules (as done by (Carral et al. 2019)) to solve class retrieval.

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Evaluation: Class Retrieval

sec

1.7M 3.1M 4.4M

10⁰ 10¹ 10² 10³

Number of assertions

Reactome

sec

1.9M 4M 5.9M

10⁰ 10¹ 10² 10³

Number of assertions

UOBM

Experimental results for class retrieval for VLog (pink/grey) and Konclude (black) Remark. Presented in (Carral et al. 2019).

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Conclusions and Future Work

Remark. We can use VLog to solve (ExpTime-)hard problems!

Future work:

• Rulewerk Extension: translate Datalog(S) to existential rules

• VLog Extension: native support for Datalog(S)

• Implement existing calculi using our approach

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References I

Carral, David, Irina Dragoste, and Sebastian Rudolph (2020). “Reasoner = Logical Calculus + Rule Engine”.In:Künstliche Intell.

Carral, David et al. (2019). “Chasing Sets: How to Use Existential Rules for Expressive Reasoning”.In:P. of the 28th Int. Joint Conf. on AI (IJCAI 2019).

Kazakov, Yevgeny (2009). “Consequence-Driven Reasoning for Horn-SH IQ Ontologies”.In:P. of the 21st Int. Joint Conf. on AI (IJCAI 2009).

Krötzsch, Markus, Maximilian Marx, and Sebastian Rudolph (2019). “The Power of the Terminating Chase (Invited Talk)”.In:P. of the 22nd Int. Conf. on Database Theory (ICDT) 2019).

Krötzsch, Markus, Sebastian Rudolph, and Pascal Hitzler (2013). “Complexities of Horn Description Logics”.In:ACM Trans. Comput. Log.

PRACTICAL USES OF EXISTENTIAL RULES IN KNOWLEDGE REPRESENTATION